##
Total Central Curvature of Curves

by Thomas Banchoff

Consider a space curve conatined in a sphere. The *total
central curvature* is calculated by taking the arithmetic mean
of the total absolute curvatures of the curves obtained by central
projection from all the points on the bounding sphere.

The aim of
this paper is to prove that this definition is equivalent
to the classical total absolute curvature of the
original space curve. This result is then generalized to
curves in
*n*-space. As a corollary, it is shown
that a curve on S^3 in
E^4 with total absolute curvature < 4 in E^4 can
be unknotted in S^3.